CKMath Unit 1: Rigid Transformations and Congruence

Focus:
To begin grade 8, students extend their prior knowledge about geometry and geometric measurement. Doing this draws on abilities developed in earlier grades to identify, compose, and decompose shapes, as well as to extend their knowledge to plane figures with different rotation and mirror orientation. Students build on these abilities and their knowledge of geometry to study transformed plane figures. They identify and describe translations, rotations, and reflections, and sequences of these. In describing images of figures on and off square grids and the coordinate plane, students use the terms “corresponding points”, “corresponding sides”, and “image”.

Students learn that angles and distances are preserved by any sequence of rigid transformations and develop a working definition of “congruence”. They experimentally verify the properties of transformations and develop informal arguments showing that alternate interior angles of parallel lines intersected by the same line are equal and that the sum of the angles in a triangle is 180°. This information is applied in later units, including Grade 8 Unit 2: Dilations, Similarity, and Introducing Slope.

Number of Lessons: 17

Instruction Time:
Approximately 45 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Focus:
To begin grade 7, students extend their prior knowledge about geometry and geometric measurement. Doing this draws on abilities developed in earlier grades to identify, compose, and decompose shapes, as well as to extend their knowledge of length and area measurements. Students build on these abilities and their knowledge of geometry to study scaled copies of pictures and plane figures. They move on to apply this knowledge further to scale drawings, e.g. maps, scale figures/models, and floorplans. This unit specifically provides the learner with geometric preparation for later learning, for example, Grade 7 Unit 2 Introducing Proportional Relationships and Grade 8 Unit 2 Dilations and Similarity.

Throughout the unit, students discuss their mathematical ideas and respond to the ideas of others.

Number of Lessons: 13

Instruction Time:
Approximately 45 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Focus:
To begin grade 6, students extend their prior knowledge about area to include shapes that are composed of more than just rectangles. Doing this draws on abilities developed in earlier grades to compose and decompose shapes, for example, to see a rectangle as composed of two congruent right triangles. Through activities sequenced to allow students to make sense of problems, students build on these abilities and their knowledge of area to find the areas of polygons by decomposing and rearranging them. They learn strategies for finding areas of parallelograms and triangles, and use regularity in repeated reasoning to develop formulas for these areas, using geometric properties to justify the correctness of these formulas.

They then use these formulas to solve problems. They build an understanding that any polygon can be decomposed into triangles, and use this knowledge to find areas of polygons. Students then apply this understanding to find the surface areas of polyhedra with triangular and rectangular surfaces. They study, assemble, and draw nets for polyhedra and use nets to determine surface areas. Students will also, throughout this unit, discuss their mathematical ideas and respond to the ideas of others.

Number of Lessons: 19

Instruction Time:
Approximately 50 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Focus:
This unit introduces students to the concept of volume by building on their understanding of area and multiplication. Students learn that the volume of a solid figure is the number of unit cubes that fill it without gaps or overlaps. First, they measure volume by counting unit cubes and observe its additive nature. They also learn that different solid figures can have the same volume. Next, they shift their focus to right rectangular prisms: building them using unit cubes, analyzing their structure, and finding their volume. They write numerical expressions to represent their reasoning strategies and work with increasingly abstract representations of prisms. Later, students generalize that the volume of a rectangular prism can be found by multiplying its side measurements, or by multiplying the area of the base and its height. As they analyze, write, and evaluate different expressions that represent the volume of the same prism, students revisit familiar properties of operations from earlier grades. Students apply these understandings to find the volume of solid figures composed of two non-overlapping rectangular prisms and solve real-world problems involving such figures. In doing so, they also progress from using cubes to using standard units to measure volume.

Number of Lessons: 11 and 1 optional

Instruction Time:
Approximately 60 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Focus:
In this unit, students extend their knowledge of multiplication, division, and the area of a rectangle to deepen their understanding of factors and to learn about multiples. Students return to the concept of area to make sense of factors and multiples of numbers. Given a rectangle with a particular area, students find as many pairs of whole-number side lengths as they can. They make sense of those side lengths as factor pairs of the whole-number area, and the area as a multiple of each side length. Students also learn that a number can be classified as prime or composite based on the number of factor pairs it has. Throughout the unit, students encounter various contexts related to school, gatherings, and celebrations. They are intended to invite conversations about students’ lives and experiences. Consider them as opportunities to learn about students as individuals, to foster a positive learning community, and to shape each lesson based on insights about students.

Number of Lessons: 6 and 2 optional

Instruction Time:
Approximately 60 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Focus:
In this unit, students interpret and represent data on scaled picture graphs and scaled bar graphs. Then, they learn the concept of multiplication. Students explore scaled picture graphs and bar graphs as an entry point for learning about equal-size groups and multiplication. The idea that one picture can represent multiple objects helps to introduce the idea of equal-size groups. Students learn that multiplication can mean finding the total number of objects in groups of objects each, and can be represented by a x b. They then relate the idea of equal groups and the expression a x b to the rows and columns of an array. In working with arrays, students begin to notice the commutative property of multiplication. In all cases, students make sense of the meaning of multiplication expressions before finding their value, and before writing equations that relate two factors and a product. Later in the unit, students see situations in which the total number of objects is known but either the number of groups or the size of each group is not known. Problems with a missing factor offer students a preview to division.

Number of Lessons: 20 and 1 optional

Instruction Time:
Approximately 60 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Focus:
In this unit, students begin the year-long work to develop fluency with sums and differences within 20, building on concepts of addition and subtraction from grade 1. They learn new ways to represent and solve problems involving addition, subtraction, and categorical data. Students are introduced to picture graphs and bar graphs as a way to represent categorical data. They ask and answer questions about situations described by the data. The structure of the bar graphs paves the way for a new representation, the tape diagram. Students learn that tape diagrams can be used to represent and make sense of problems involving the comparison of two quantities. The diagrams also help to deepen students’ understanding of the relationship between addition and subtraction.

The unit is also designed to give students time to learn the structures and routines for centers, to create norms for classroom learning, and to begin to build a mathematical community.

Number of Lessons: 14 and 4 optional

Instruction Time:
Approximately 60 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Additional Search Terms:
picture graph • bar graph • categorial data • tape diagram

Focus:
In this unit, students deepen their understanding of addition and subtraction within 10, and extend what they know about organizing objects into categories and representing the quantities.

The activities in this unit reinforce addition and subtraction word problems within 10 using objects and drawings, and initiate the year-long work of developing fluency with sums and differences within 10. Some problems involve finding sums greater than 10, a skill to be honed throughout the course and with the support of tools such as connecting cubes. Students use drawings, symbols, tally marks, and numbers to represent categorical data. They go further by choosing their own categories, interpreting representations with up to three categories, and asking and answering questions about the data.

The unit is also designed to give students time to learn the structures and routines for centers, to create norms for classroom learning, and to build a mathematical community.

Number of Lessons: 14 and 1 optional

Instruction Time:
Approximately 60 minutes
(Please note that each lesson is designed for one instructional block, and may be divided into shorter or longer segments based on teacher pacing and student needs.)

Additional Search Terms:
sort • survey • representation • record • organize • tally • expression