## An update on our new content for the United States (Jan to Mar 2018)

#### Elementary

The distributive property for multiplication is introduced to students in Grade 3 through the idea of the area model for multiplication. Students identify that the area of 9 x 27 is the same as the area for 9 x 20 and 9 x 7 added together. This becomes an effective strategy both for multiplication, as well as a foundational skill for applying the distributive property to algebraic expressions in middle and high school. Two subtopics of questions were set live this month related to these skills. Distributive property for multiplicationand Use the distributive property.

Also in this area, students start to apply the idea of an unknown to multiplicative statements. There are 3 new subtopics this month to enable familiarity with this idea. The first one utilizes students’ basic facts, and combines this with notations involving variables. The next 2 focus on 4 operations and on combining operations together (for Grade 4).

In measurement a great lesson went live on Shapes and fractions — perimeter. This would be an extension opportunity for 4th grade students. The video here guides students to proportional thinking in identifying strategies for finding the fractions around the outside of a shape. This is based on an activity I used to run with my students in the classroom, starting easy like finding one third of the way around a triangle, up to harder ones like finding ⅕ of the way around a triangle. It’s a great activity to run with your class and encourage class discussion on their thinking and understanding.

Operations and Algebraic Thinking in Grade 4 moves to generate rules to describe patterns and relationships. Additionally, students begin to understand decimal notation for fractions in the tenths and hundredths. We have 4 new sets of work to help students synthesize work from these clusters by analyzing number patterns for decimals and fractions in the tenths and hundredths. We have scaffolded the work for decimal patterns in tenthsfollowed by hundredths, and then fractions using tenths or hundredths and then finally a last set that combines this knowledge together.

Another example of how at Mathspace we try to combine ideas and knowledge together is in our new 4th grade lesson on Addition and subtraction problems involving angles. Found in the whole number addition and subtraction topic, this subtopic combines skills of adding 2 and 3 digit numbers together with the newly acquired understandings of putting a measure to a rotation and calling it an angle. This lesson has a number of simple geogebra applets to help consolidate this understanding.

Also applicable to this age group are the following new lessons in Measurement. All of them with explanatory videos for students to watch, or for teachers to use as stimulus for discussion.

- Shapes and fractions with area (Grade 3)
- Comparisons using volume (Grade 5)
- Measure, estimate, order and compare objects (Grade 5)
- Comparisons using area (Grade 3)
- Areas of Regular and Irregular Shapes (Grade 3)

#### Middle School

Long division is necessary in High School for use with Polynomials, and it is required in Grade 7 to look at long division to convert a rational number to a decimal. So leading up to this are the skills required to perform long division on numbers. We have scaffolded a set of long division work to assist in this area.

Long division for 1 or 2 digit numbers (resulting in decimal)

Long division for 3 digit numbers (resulting in decimal)

Long division for 4 digit numbers (resulting in decimal)

Long division — divide decimals by 1 digit numbers

Long division — divide decimals by 2 digit numbers

While not new this month, I thought I would highlight a number of excellent lessons in Middle School Measurement.

Area of two dimensional shapes, whilst starting in elementary with the understanding of rectangular areas, is extended throughout high school. The fundamental underpinning is the understanding of the area of a rectangle: A = l x w. Once students really understand the array structures that support this, we take them on a journey to connect the area of other shapes back to the area of the rectangle. Most of these lessons have simple geogebra applets to demonstrate the connection

Triangles (half of the rectangle)

Parallelograms (almost a rectangle)

Trapezoids(similar to parallelograms, which are almost a rectangle!)

Kites (half of a rectangle)

Rhombus (yep, just like a rectangle!)

Circle (yes, we even relate a circle back to a parallelogram — which is like a rectangle!)

#### High School

Have you seen our coverage of complex numbers? 28 lessons in the ebook so far, most with interactive questions. From basic operations of complex numbers, up to sketching circles and ellipses on the complex plane. This month the subtopic Factoring and Expanding Complex Numbers was set rounding out our basic operations of complex numbers.

Vectors in Mathspace is something that we are working hard on to bring you as fully interactive content. At the moment however, you can find amazing coverage of 46 lessons relating to Vectors in our eBook.

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