Understanding Perfect Cubes and cube Roots
Lesson 3 of 19
Objective: Students will be able to use a visual understanding to make connections to area and volume as applications of squares, square roots, and cubes, cube roots
After completing the perfect squares activity using the square tiles to model values squared visually, completing this perfect cubes activity should run much more smoothly. This activity is meant as the follow-up to perfect squares and extends the conceptual understanding already established with squares into cubes and also cube roots. I wanted to take full advantage of this visual modeling by asking students to make connections to perimeter, area, and volume. The extension activity asks students to apply squares, square roots, cubes, and cube roots to geometry application using perimeter, area, and volume. The extension also begins to move students towards graphing and looking for patterns in the graph. Graphing correctly is a major focus of eighth grade math as is looking for patterns such as liner (perimeter), quadratic (area), and cubic (volume). Why do we call some patterns linear and what do they look like graphically so students understand how to identify non-linear patterns? Of course students will study linear patterns more closely in later units as well.
- Print a set of the square and square root cards. Copy and cut enough cards so that each student has a card of their own.
- Prepare a number line for a wall of your classroom. Number lines can be purchased at teacher stores or made using thin rope from Lowe’s and index cards for numbers. Cloths pens or paper clips either make good hanging devices. Your number line will not need to include negative numbers but will need to extend up to at least 225 if you use the 152 card.
There are two parts to this lesson opening activity today. Part one involves graphing the value of squares and square roots on a number line while the second part involves a formative assessment piece. As students enter the room, hand them a pre-printed and cut expression card with either a value squared or the square root of a value. Inform students they need to graph the value of their card on the given number line across the room estimating when necessary to the best of their ability. Move about the room assessing student work and providing feedback to move student learning forward.
Once each student has graphed their expression, pull the class together for a mini-wrap up discussion of the number line. During the whole class wrap-up, discuss such questions as, “Why is the square root of 20 located between two numbers? What is happening to the value of the radical as the number under the radical is growing larger? Why are so many of these radical cards located between numbers instead of on an exact number? Why aren’t any expressions graphed on the negative side of the number line?
After discussing the graph, put the exit slip questions on the board and have students answer these completely on their own as a formative assessment. I usually cut scrap paper into fourths and offer it as bellringer or assessment paper. These formative assessment questions are designed to reveal students’ conceptual understanding of squares and square roots as well as their ability to perform mathematical operations using these symbols. Collect the assessments and review for possible regrouping of cooperative groups homogenously or for intervention and enrichment services throughout the unit.
Cut or purchase square tiles that are either one inch square or one centimeter square. I use craft foam from the Dollar Tree and a die cut to create many tiles in a variety of colors.
Gather any type of small cubes you have into small containers or bags for each group. I usually borrow small squares kept in little round plastic containers. Our math consultant for the districts has classroom sets of every manipulative imaginable. Children’s toys and wooden blocks are also a option for cubes. Basically, any type of cube very small to moderately large is fine as long as you can get enough sets for each cooperative group to have a container of at least 100.
Beginning the Lesson
Clarify for students that today is focused on extending their current understanding of Perfect Squares to include a visual understanding of Perfect Cubes. Write the following expression on the board, 52, and ask a student to read the expression out loud and explain what it means. As the student is talking and saying important information that will help students as they work through the activity today, script their words on the board next to the expression.
Write the following expression on the board, 43 and tell students that we read this expression as “four cubed.” Ask for a volunteer to offer an opinion on what is meant by 4 cubed and script any guesses on the board for consideration. Tell the class that today we will work through a hands-on activity designed to help students answer the question, “What does it mean to cube a number?” The activity will also include a few extensions when we are finished.
Group students into cooperative groups of two to three students and give each group a bag or handful of square tiles along with a container of cubes. Give every student a copy of the handout Perfect Cubes Activity.
Instruct students to work within cooperative groups to complete the table. The table asks students to construct models in both 2-D and 3-D then sketch them in the table. Remind students they are to actually build the squares and cubes first then sketch them and answer the other columns. Once they complete the table, groups should call you over to check their progress before moving on to answer the questions.
As students work within groups and provide feedback to each other, walk about the room providing feedback that also moves learning forward. I usually begin with the groups that I know have less prior experience with exponents to ensure they are beginning productively. After checking on all the groups who may struggle, I check with the groups who showed more prior knowledge to ensure they are also working productively. As you provide feedback to groups make note of different approaches to the answering the questions and begin to ask groups to present ideas to specific questions during the mini-wrap up session. I like to tell groups ahead of time which questions they will present and what thinking I really like from their work so that they are ready to speak and they know what it is I really want them to mention. I also carry my ipad with me (any picture taking device will work) to take pictures of the models as students use the tiles and cubes to construct perfect squares and cubes. I like to post these pictures to my teacher website or Edmodo.com so that students can use them as resources for studying and I can refer back to these images during classroom discussions throughout the unit. After you have teacher checked most groups’ tables, pull the class together for a very quick mini-wrap up of the table and put a student paper under your document camera to discuss answers. You could also put your ipad under the document camera and use it to discuss the differences in perfect squares and perfect cubes. Some good discussion questions could be, “What is the difference between a square and cube? (looking for answers such as square is 2-D and cube is 3-D or the cube is not just building length and width but also height.) “How is this activity different from the perfect squares activity from yesterday.” (Again looking for height and that if the dimensions were 4, then the stack needed to be four levels tall).
Once you complete the first mini wrap-up, ask students to work together in cooperative groups to answer the follow-up questions in question 1 parts a-e up to the extension activity. If your class period ends before students finish this section do not assign these questions for homework as they are designed to generate some good discussion among cooperative groups. The value in these questions is in the discussion that is created among group members. You may not have time to even begin these questions on the first day and that is fine. Leave these questions for class time on the second day of the activity instead of working alone for homework.