The purpose of this activity is to elicit ideas students have about connecting cubes. Students learn the
Notice and Wonder routine which will be used throughout the year. This routine provides an
opportunity for all students to contribute to the conversation and for the teacher to listen to what
knowledge students already have.
For all of the routines, consider establishing a small, discreet hand signal that students can display to
indicate they have an answer they can support with reasoning. This signal could be a thumbs-up, a
certain number of fingers that tells the number of responses they have, or another subtle signal. This is
a quick way to see if students have had enough time to think about the problem. It also keeps students
from being distracted or rushed by hands being raised around the class.
A picture of connecting cubes is provided. However, it is preferable to display a collection of actual
connecting cubes. Students will also have a connecting cube to examine up close.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important aspects of the task and its potential use.

This is the first version of a task asking students to find the areas of triangles that have the same base and height, and is the most concrete.

This is the second version of a task asking students to find the areas of triangles that have the same base and height.

The purpose of this task is two-fold. One is to provide students with a multi-step problem involving volume. The other is to give them a chance to discuss the difference between exact calculations and their meaning in a context.

This is the third in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. Here, we are given the volume and are asked to find the height. In order to do this, students must know that 1 ℓ=1000 cm3. This fact may or may not need to be included in the problem, depending on students’ familiarity with the units.

This is the last in a series of four tasks that gradually build in complexity. The purpose of this series of tasks is to build in a natural way from accessible, concrete problems involving volume to a more abstract understanding of volume. This problem is based on Archimedes’ Principle that the volume of an immersed object is equivalent to the volume of the displaced water. While the stone itself is an irregular solid, relating it to the displaced water in a rectangular tank means that the actual volume calculation is that of a rectangular prism, and therefore, fits in with content standard 6.G.2.

The purpose of this task is for students to translate between information provided on a map that is drawn to scale and the distance between two cities represented on the map.

The purpose of this task is to have students explore various cross sections of a cube and use precise language to describe the shape of the resulting faces.

The purpose of this task is for students to solve a volume problem in a modeling context (MP.4).

This goal of this task is to give students familiarity using the formula for the area of a circle while also addressing measurement error and addresses both 7.G.4 and 7.RP.3.

The goal of this task is to motivate and prepare students for the formal definition of dilations and similarity transformations.

This task has two goals: first to develop student understanding of rigid motions in the context of demonstrating congruence. Secondly, student knowledge of reflections is refined by considering the notion of orientation

This task shows the congruence of two triangles in a particular geometric context arising by cutting a rectangle in half along the diagonal.

The purpose of this task is for students to study the impact of dilations on different measurements: segment lengths, area, and angle measure.

The purpose of this task is to apply rigid motions and dilations to show that triangles are similar.

The purpose of this task is for students to measure angles and decide whether the triangles are right or not. Students should already understand concepts of angle measurement (4.MD.5) and know how to measure angles using a protractor (4.MD.6) before working on this task. Students should also understand that a defining attribute for a right triangle is that one of the three angles is 90 degrees. The task asks students to choose the appropriate tool which gives them a chance to decide for themselves what they will need to answer the question, thus allowing them to use appropriate tools strategically (MP5). The triangles in this task are smaller than most classroom protractors, so students have several choices. They might extend the sides of the triangle using a ruler. It might help them to cut out the shapes and place them on another piece of paper. A second and slightly more difficult method is to leave the polygons on the page and list the letters for the polygons in the appropriate entries of the table. For this students will need to manipulate the protractor or the orientation of the paper instead of rotating the shape itself, which can also serve a useful purpose later on. Alternatively, students might use the corner of a piece of paper that they have measured and know is 90 degrees, and then compare the angle formed by the edges of the paper with the angles in the triangles. The attached black-line master is for the version that encourages students to cut out and manipulate the polygons in ways that might make it easier to measure. An extension of this activity could be to have students classify the triangles in the right side of the chart based on other defining attributes (scalene, isosceles, equilateral, obtuse, acute).

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, only one practice connection will be discussed in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.

This particular task engages students in Mathematical Practice Standard 5, Use appropriate tools strategically. The task asks students to choose the appropriate tool which gives them a chance to decide for themselves what they will need to answer the question, thus allowing them to select appropriate tools. They may directly measure the angles of the triangles or may use a 90 degree template that they have constructed. As students become proficient in this practice, they will be able to consider a tool’s usefulness and consider its strengths and limitations, as well as know how to use it appropriately. The solution pathway that a student selects needs to make sense to him/her to be able to explain and justify (MP.3). Students will come to realize that certain methods/tools are more efficient and they will abandon less useful tools in favor of more appropriate strategies/tools.

This task was created as part of an Illustrative Mathematics Web Jam.

This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: First pose the question: Here are four triangles. What do all of these triangles have in common? What makes them different from the figures that are no...

Overview: This is a task from the Illustrative Mathematics website that is one part of a complete illustration of the standard to which it is aligned. Each task has at least one solution and some commentary that addresses important asects of the task and its potential use. Here are the first few lines of the commentary for this task: Materials * A copy of Grandfather Tang's Story by Ann Tompert * One set of tangrams for each student (see note in commentary) * A set of tangrams for t...
Subject: Mathematics

The purpose of this task is for students to represent and interpret categorical data. In first grade, a bar graph is a bit advanced, but the task itself is on the easy end for second grade. So this task could be used with advanced first graders or second graders just beginning to work with bar graphs.