

BASIC THINKING MATH  LEVEL 17 

Each month, we will highlight some of the key learning goals, teaching tips, and additional activities you can do with your students enrolled in the math program. We will provide specific instructional techniques on various levels to help you lead our students to master the key math concepts. 

PRIMARY GOALS: 

• 
Booklets 113: Understand fractions (how to convert between proper fractions, improper fractions, and mixed numbers); add and subtract fractions with the same denominator.


• 
Booklets 1418: Know how to find the perimeter and area of common shapes (e.g. triangle, square, rectangle, rhombus, trapezoid) and irregular shapes (a combination of the basic shapes).


• 
Standard Competition Time: 1.5  2.5 minutes



KEY TEACHING TIPS: 

• 
Mastery: Students must fully understand the basic concepts of fractions, as these are important for learning to combine fractions with same or different denominators. Understand the origin and then be familiar with how to use all basic formulas for finding the area of common shapes.






1. In Level 17, students are not required to simplify fractions. (If they do simplify, do not mark it as a mistake.) From Level 18, once they learned how to simplify fractions, all fractions must be simplified to the lowest term. At this point, if a fraction is not simplified, it should be marked wrong.




2. Be aware of the different definitions of improper fraction that a student may encounter in school. Typically (and in Eye Level), an improper fraction is defined as one in which the numerator is larger than the denominator. Sometimes, the definition will be extended to include fractions that have the same numerator and denominator (e.g. 7/7, 123/123). So, when grading a student’s work, if a number such as 3/3 is provided as improper fraction, let the student explain the answer, as the school may have presented that as an improper fraction.



• 
Level 17, Booklets 1418: This section progresses quickly. Each booklet usually consists of at least two formulas to find the area of selected shapes. Students should be coached to try both equations during class. In Booklet 18, students calculate the area of shapes that are formed by a combination of basic shapes (e.g. squares, rectangles, triangles, trapezoids). Therefore, it is important to ensure that students fully understand the formulas for the area of those basic shapes before assigning Booklet 18, so that students are confident to perform the required calculations once they learn how to divide the irregular shape into the basic shapes.



• 
The Meaning Behind the Area Formulas 1. Square/Rectangle: A = L × W
Area is the total number of unit squares in each row times the number of rows.



• 
2. Triangle: A = ( B × H ) ÷ 2
Area is half of the area of a rectangle.



• 
3. Trapezoid: A = ( B1 + B2 ) × H ÷ 2
When two identical trapezoids are combined, they form a parallelogram, and the area is more easily calculated. Then, divide the area of the parallelogram by 2 to get the area of the trapezoid.



• 
4. Rhombus: A = D1 × D2 ÷ 2
When the rhombus is divided diagonally into 4 equal portions as shown, the pieces can be rearranged to form a rectangle which has half of D1 times D2.




Coaching Notes: 

• 
Relevance:
Help the student to relate fractions to real life examples or situations. Share these examples, then ask the student to provide similar ones.



1. How are mixed numbers represented in real life?




If the bakery makes several identical cakes (of any shape), and each one is cut
into 8 equal pieces, then means that there are 2 full cakes and 3 additional pieces.



2. How are mixed numbers and improper fractions represented in real life?




If there are 19 pieces of cake, and it is known that each cake was cut into eight pieces, then the pieces could be reassembled to form 2 cakes with 3 pieces left.



3. How is addition of fractions with the same denominator represented in real life?




If all cakes (of the same type, size, and shape) are cut into 8 pieces, then the above expression can be understood by considering whole cakes and pieces separately.
For example, 1 whole cake plus 2 whole cakes PLUS 3 pieces plus 7 pieces TOTALS 3 cakes and 10 pieces. So, the answer is 3 10/8. Since each cake has 8 pieces, from 8 of the 10 pieces another cake is formed, with 2 pieces leftover, leaving 4 2/8 cakes.




Application/Challenge:
Make up a story, and ask the student related questions that require them to utilize calculations with improper fractions.




Example: Jane is having a party, and she wants to buy 11 pieces of cake from a bakery that cuts all of its cakes into 6 pieces each. The bakery has 3 cakes and 1 additional piece. Ask the student:




(1) How is this situation represented with an expression that uses improper fractions?





(2)Are there enough pieces for Jane to buy?



Answer: Yes.




If Jane buys 11 pieces, how many pieces/cakes will be left?
Answer: 1 full cake and 5 pieces.




(This scenario requires students to construct, compare, convert, borrow, and reduce.) 


• 
Reinforcement: Let students copy the concepts and explanations of each formula into the Key&Note. This reinforces how formulas of various areas are derived and give students a better understanding when working with area. Encourage students to be very neat when drawing pictures so that visuals are accurate and helpful.



• 
Common Problems:
1. The concepts of base and height are often a stumbling point for students. Make sure students know that the height is represented by a line extending perpendicularly (i.e. at a right angle) from the base to the top of the shape.




Ask students to refer to the images above as they answer these questions:
• Which letter represents the base? [B][B1,B2]
• Which letter represents the height? [C]
Note: Students often think that the base is the line at the bottom (D) and height is the vertical one line (B/B1/B2).
Draw similar shapes in the Key&Note and/or refer to shapes within the booklets to practice further and to confirm understanding.
2. Some questions give the area of the shape and ask the student to find the height or the base/diagonal. This requires the student to be very familiar with the formula and able to work backward to find the answer.
Coaching Tip: Ask the student to recall the formula before doing the work.
3. Some students will overlook the details and just apply the formula.
Example: When asked to calculate the area of the figure shown here, the student may calculate 4 x 5 ÷ 2 = 10. (They do not realize that the numbers there are not the actual diagonal line.) How do you coach this student?


