# Translations

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## Objective

SWBAT describe and perform translations on objects in the coordinate plane

#### Big Idea

Students will follow Alice down the rabbit hole to explore mathematical translations.

## Do Now

7 minutes

As students walk in the room, I hand them the Translations Do Now which has a grid with several triangles plotted on it. Although this activity is similar to the Do Now from my lesson on reflections, students are given a different grid and quote. Students are instructed to describe the translations that map triangles onto other triangles. This activity is designed to identify students’ prior knowledge about translating objects on the coordinate plane. It leads into the lesson where students will use functions to describe translations (G.CO.2).

An additional question asks students to explain how a quote relates to transformations. The quote, from Alice’s Adventures in Wonderland,  by Lewis Carroll, said by Alice, is about falling down the rabbit hole and falling off the roof of her house. I like to use this quote to add some literacy and real-world connections to the lesson. Students recognize the quote as relating to translations which shift objects along the plane.

For more information about how and why I used Lewis Carroll’s writing in my classroom, see my video “Lewis Carroll and Mathematics.”

Intervention Resource: Working with translation rules requires students to have fluency with adding and subtracting integers. Since many of my students have difficulty with this or may be out of practice, I often add some practice problems to identify their competency. I have included a modified Do Now, Translations Do Now with Integers  that has a question which addresses adding and subtracting integers.

## Mini-Lesson

20 minutes

As the students are working on the Do Now, I hand out the “Translations Mini-Lesson” worksheet. We begin the Mini-Lesson by going over the Do Now. Based on the discussion of their answers, I can see what students’ level of prior knowledge is. Most students are aware that movement along the coordinate plane cannot be in a diagonal direction. However, many students are unsure whether objects first move vertically or horizontally. We move into a discussion on translation vectors and how they differ from coordinates. I ask students to define the term “coordinates” and to think about how a vector might differ. They fill in the blanks of the statement, “Coordinates show … and vectors show….” I look for a response, such as, “Coordinates show where a point is located and vectors show how a point moves on the coordinate plane.” I have the students write the vector, and we discuss the meaning of the signs of the x and y.

Looking at vectors helps students understand how point move when they are translated. This moves the discussion towards writing and analyzing rules. We look at how “input,” “output,” and “function” can be related to “pre-image,” “image,” and “translation” (G.CO.2). When translating an object, I advise students to look at one specific point in the object rather than the whole object. I find that the most common mistake students make is moving a point on the pre-image to a non-corresponding point on the image.

In this part of the guided practice, students will describe the translation of a triangle in four different ways. They will refer to the graph from the Do Now. It is also helpful to project the graph on the board. In the “Description” column, students write a description of the translation in words, i.e.,”three right and one up.” Then students write the translation as a vector. In the last two columns, students write the rule in translation notation. I have them choose and label a point on Triangle A and uses its coordinates to write the rules. After we do the first row together, I have the students work with the person next to them to complete the other three rows. We go over the chart and I address and misconceptions, such as mixing up the notations or the order of the x and y coordinates.

The last part of the guided practice shows the students three types of questions they may be asked to answer using coordinates only. At this level, students should be able to interpret rules and perform translations without using a graph.

## Activity

13 minutes

After the guided practice, students should be able to work on the practice sheet independently. I circulate around the room as students work. However, I only answer clarifying questions mostly to do with correctly interpreting the prime notation. Most of the students other questions can be answered by referring back to their guided practice sheet or by asking another student. Sometimes I will pull a group of students together who are struggling to reteach the mini-lesson.

After about 10 minutes, we go over the answers to the sheet. I ask a volunteer to show his/her graph on the document camera and then ask other students to give their responses from questions 2 and 3.

## Summary

5 minutes

For the Exit Ticket, students are given a false translation rule and asked to explain why it is false and how it can be changed to become true. I like this question because there are several possible changes that can be made. The coordinates of the pre-image or image can be changed or the translation vector can be changed.  If there is time, we discuss the answer. On their way out, I collect the students’ sheets to assess their understanding of the content.