ELD 374 Midterm

procedural understanding

follow the steps and use the rules to complete the problem, dont know why you do what you

½ + ¼ =

conceptual understanding

being able to solve problems with understadning

number blocks, ten frames, representing 65 + 90 with tens and ones blocks

*give diff ways, focus on this

School principles and standards for mathematics are defined by…

National Council of Teachers of Mathematics (NCTM).

What are the five content standards?

• what do you teach

• number/operation

• algebra

• geometry

• measurement

• prob. + statistics

what are five process standards?

• how do you teach

• problem solving—26+35, maddie had 35 dollars, sam had 26 dollars how many in total, word problems

• representation—-drawings, geo boards, how many times greater is area of square when you double the size, then using drawings

• connection— explain how different math concepts connect, bring real life into math,1/2= 0.5=50%,

• communication— 2/4, numeration + denominator, fractions are a part and whole

• reasoning/proof —explaining why and how, is a sqaure is always a paralleogram, bc opp. sides are congruent and parallel

how do we derive the area of triangle from rectangle?

• 1/2bh bc its ½ of a trianle

principles of school math

• teaching

• learning

• assesment

• cirriculum

• tech.

• equity

Eight Standards for Mathematical Practices

1. Make sense of problems and persevere in solving them.

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

guide educators on the different types of expertise students should develop, what you should focus on when constructing a lesson

behaviorism

• watson

• drill + practice, memorization, not focused on conceptual understanding, reward + punishment

• learning by repetition

constructivism

learning is active meaning seeking process, learners are not passive consumers of information, focus more on mathematical thinking rather than correct answer, let students discover math concepts

negotiation, explanation, sharing, evaluation

piaget

zpd

Zone of Proximal Development (ZPD) is the sweet spot between what a learner can do all by themselves and what they can do with help

• child knows addition, but not multiplication, provide help with multiplication to learn division

CRA approach

concrete (inactive) level—-actions, hands on learning, having students measure the angle, learn by doing, anglex

representation (iconic) level—-visual representation, drawing, having students draw parallegram and two trapezoids to show their relation with area

abstract level (symbolic) —equations—using abstract symbols, 4+4=8, calcuating area using formula, develop math ideas

scaffolding

guidance

sociocultural theory

learning is not isolated from your social cultural environment, they influence your learning

big ideas of geometry

shapes and properties— 4 angles, 4 sides

transformation—move smthing, make smthing bigger or smaller, reflection, rotation, dilation, translation

visualizations—plane and space

location— we can locate objects with some reference, normally on an axis,

plane vs space

plane— 2d, space—3d

polygon vs non polygon

polygon—- closed shape that was formed by joining 3 or more straight lines, multi side

non polygon—a closed figure, shape had curves, intersecting lines,

concave vs convex

convex— all interior angles less than 180 degrees, and at least one diagonal lies inside the polygon, both regular and irregular

concave—at least one interior angle is greater than180 degrees, and at least one diagonal lies outside the polygon, never regular

which ones with these are concave and convex

The polygons that are concave are B, E, H, I, and L. The polygons that are convex are A, C, D, F, and G

irregular vs regular polygons

Regular- when each side and as well as each interior angle are equal, triangle, square, pentagon

Irregular— when angles and sides are not equal, rectangle

Van Level

• 0: Visualization— what students can see with the naked eye, blue angle

• Level 1: Analysis—know triangle has 3 sides, 3 angles, if traingle is presented diff. they may say left angle, know properties

• Level 2: Informal deduction—know sqaure is a rectangle

• Level 3:Deduction—know proofs and theorems, highschool level

• Level 4: Rigor— Develop logical mathematical thinking, college level math

** Provide a description of how students progress through levels of geometry

types of triangles

scalene— all sides are not equal

isosceles- at least 2 sides are congruent, equilateral is isos.

equilateral— all sides are congruent,

actue angle— all angles are less than 90

right angle— one is angle is 90

obtuse--only one angle is more than 90

obtuse

obtuse— never acute, right, equilateral

smtimes- isosceles, scalene

quad. and their properties

square— all sides are congruent, all angles are 90 degrees, opp. sides are parallel and congruent

rectangle- opp sides are congruent and parallel to each other, all angles are 90

parallelogram—opp sides are congruent and parallel to each other

rhombus—all sides are congruent, opp sides parallel to each other

trapezoid—one pair of sides are parallel

kite— 2 sides adj. sides same length, equal to each other

**square can be any of these

** at least set of opp. sides are parallel for all!!

** diagonals bisect each other expect for kite/trapezoid

** diagonals congruent/ all right angles only for sqaure,rectangle (or isos. trapezoid)

**diagonals interect at right angles only for rhombus, square, kite

**opp sides not congruent only for trapezoid, kite!!

area for quad.

sqaure, rectangle rhombus, paralleogram— all l x w

trapezoid- 1/2( b1 + b2) H

kite— (d1 x d2)/2

**for rhombus/ kite also ½ of area of rectangle

** for kite, break down into 2 rectangles divide by 2 since they are really triangles, then add the 2

Explain the how does area of rectangle, parallelogram, and rhombus have the same formula to find the area. Explain by using suitable diagrams

The rectangle, parallelogram, and rhombus have the same formula (l x w) to find the area because the rhombus and parallelogram can be rearranged into a standard rectangle without changing their area, and all three formulas are based on multiplying a base by the perpendicular height.

Regarding the parallelogram, you can form a triangle along its perpendicular height and move it to the other side to form a rectangle.

For the rhombus, if you divide it into two triangles and then add the two areas together, you will obtain the total area. Since the area of each triangle is ½ (b x h), adding the two areas results in b x h. Therefore, the area of the rectangle, parallelogram, and rhombus are the same

Explain the how does formula to find the area of parallelogram can be used to derive formula to find the area of trapezoid. Explain by using suitable diagrams

The formula to find the area of the parallelogram can be used to find the area of a trapezoid because two identical trapezoids can be combined to form a parallelogram. The total base of the parallelogram is the sum of the two parallel sides of the trapezoid (base 1 + base 2), and the height remains the same. Since we are only concerned with the area of one trapezoid, we divide that total area of the parallelogram in half, and the formula for a trapezoid becomes 1/2 (base 1 + base 2) x h

sometimes, always, never with quad…

Shape	Rectangle	Square	Parallelogram	Kite	Rhombus	Trapezoid

Rectangle

Always

Sometimes

Always

Sometimes

Sometimes

Never

Square

Always

Always

Always

Always

Always

Never

Parallelogram

Sometimes

Sometimes

Always

Sometimes

Sometimes

Never

Kite

Sometimes

Sometimes

Sometimes

Always

Sometimes

Sometimes

Rhombus

Sometimes

Sometimes

Always

Always

Always

Never

Trapezoid

Never

Never

Never

Sometimes

Never

Always

**if trapezoid is at least one pair of parallel sides than all shapes are always a trapezoid,

**trapezoid is smtimes parallelogram, kite