Plotting... on a Cartesian coordinate plane

Students have taken a slight break from exploring fractions to lessons with the coordinate grid. Reading, writing, and plotting points on the first quadrant requires practice and purpose. Students were asked to represent the axes for the first quadrant and plot four destinations using cardinal directions.  

My example is here:

Directions for students:

Student examples: 

This assignment was not only informative (I used this as one of our bodies of evidence to support a Critical Concept) but students really enjoyed it! I had many students want to make additional maps "just for fun". Assuredly, that enthusiasm does not happen often during a math assignment! I appreciated the creativity that some displayed when plotting their locations. Whether students chose "rappers" or "athletes" as the stops and final destination, the assignment provided practice and a piece of evidence demonstrating how well students are able to produce and plot ordered pairs on a coordinate grid.

 Missouri Learning Standards addressed:

Define a first quadrant Cartesian coordinate system. a.    Represent the axes as scaled perpendicular number lines that both intersect at 0, the origin. b.    Identify any point on the Cartesian coordinate plane by its ordered pair coordinates. c.     Define the first number in an ordered pair as the horizontal distance from the origin. d.    Define the second number in an ordered pair as the vertical distance from the origin.

Plot and interpret points in the first quadrant of the Cartesian coordinate plane.

Fraction Action!

 Students are knee-deep with fractions. When you say Fractions, even many adults admit that they "hate" fractions. Yet, teaching fractions has provided some of my most joyful teaching days in math!

A wonderful Colleague introduced me to this post by Graham Fletcher:

Below, you can see how students talked through the points on the number line.

I was pretty impressed. You can see that my first "B" was not really centered at the 1/2 mark (really the second one is not great either!) and students originally noted 3/8. When the student explained why it was 3/8 vs. 4/8, I knew that I needed to redraw my 1/2 mark for the rest of the conversation to move forward.

The number line was presented in purple. The annotation in pink marker shows the kids' thinking.

This string was fun! Students recognized the pattern of the half from the previous string becoming the endpoint for the next string. In other words, we kept narrowing in on the values between the half and the whole.

We have also started many lessons by thinking about how we can label fractional parts and how they are equivalent to other parts.

I highly encourage you to try these with your students as they were informative and built conceptual knowledge along the way!

Students shared their ideas:

Here is another example where students 

shared their thinking:

Original:

I loved the green outline of the quadrilateral that was 3/8. I did not see that one!

 

Division: Choose Your Own Numbers

 Students have been working on multidigit multiplication and are practicing division each day to build the relationship between the two operations. In addition to utilizing number strings, students are working together to use various strategies to solve equal share problems.

Students were presented with problems that they could solve "as is" or adjust the numbers by choosing from the other options. When students choose the numbers, this provides the teacher with information too! Are they choosing numbers that are a "good fit"? A challenge?

You can see how the student is creating a model to represent the problem. She did have three pandas and she is trying to distribute the "bamboo" equally to each bear by using tally marks. While she needed some support, the teacher can see her approach and that she is able to approach the problem independently.

This student is using a partial quotient strategy utilizing multiplication. Does his "answer" match his work? What questions could move him forward? What does his 1/3 represent? Where is that 1/3 coming from if you had a difference of zero?  

Here you can see some student strategy use:

Invasive Species!

 Students have completed our Life Science unit by investigating various invasive species. Using one of our Benchmarks reading series texts, students selected an invasive species and identified the problem and solution to reduce the spread of some challenges that are facing our environment.

Visualizing Length

 How much is 10 times more? A 1/10 of? The answer depends on what your whole is. In our opening lesson introducing powers of ten, we did an investigation using the book Actual Size  by Steve Jenkins as our anchor text.

Our first task was to determine the size of an average fifth grader in our class. Students lined up from shortest to tallest and we found the median height which was 60 inches (5 feet) and represented by the brown paper outline. 

The book's closest animal that was 1/10 of 5 feet was the Goliath Beetle. 

Measuring out the Giant Squid (59 feet in the book) we rounded to 50 feet and said that our squid was not yet full-grown to benchmark it to 10X more than 5 feet. We tried to measure the length in the classroom using meter sticks.

Using a long hallway and yarn to mark the length, 

we measured 50 feet!

Students were challenged to visualize what 10 Time MOre of the Squid would look like! We knew we would have to go outside as the commons area in our building would still be too short. 

We also discussed how if we would have started with the Goliath beetle as our whole, we could have easily determined 100 times more length but it would have been difficult to represent 1/100 the size of the Goliath Beetle.

Exploring Volume: It's More Than a Formula!

This week we have transitioned from area to volume. Students started the investigation using 3D boxes and centimeter cubes to estimate how many cubes would fill the space of the cube. This year, I removed the "lines" that traditionally are preprinted on the cubes. This removal of lines dramatically changed the types of estimates that were offered from students.  

Eventually, students were encouraged to fill the boxes with the cubes to determine the actual number of cubes needed to fill the space of the box. 

Our next investigation involved a book by Jo Boaler which you can find in the book Mindset Mathematics. Want to see the book? Find the link here. After our first investigation of filling the boxes, it was evident that students were still developing their ideas of volume and how the different views of each face can help us find the volume.  This task is not about "finding the volume" rather how we can rotate and turn the cube and making sense of the images. I compare this activity to solving a Rubik's cube and at the end discovering that you have one solo tile that is out of place. A bigger challenge than meets the eye!

  

Students will continue to finish the investigation as they progress through the volume unit. 

As we continue to experience volume, students were asked to build cubes and rectangular prisms with given dimensions (see below). This is reinforcing What is length? What is width? What does that number represent? An observation that became clear was many students believe that just rearranging the order of the dimensions could "change the size" of the prism. This opened up a conversation about what those numbers mean.  We rotated their prisms and showed how rearranging the dimensions does not change the shape's presentation. I noted that this is the commutative property of multiplication.  

 To help students see how they could "change" the dimensions without changing the volume, I returned to a number string that I had opened the lesson with. I asked how the part squared off in red could help them with problem number 3 on their worksheet? The conversation was productive and students were revising their dimensions with greater success.