Angles

[UHP]

For a few months I've been presenting the lessons in "Connecting Math Concepts Level E" to my daughter. It might be "fourth grade math" if Level A is kindergarten. We're about halfway through it.

In the last couple of lessons the program introduced angles. I thought this passage from the teacher's guide (which follows an excerpt from Lesson 72; I've reproduced the same excerpt further down) was interesting:

Quote

Teaching note: This exercise should not be difficult for the students; however, it is very informative about the subtle properties of angles. Angles can be added, subtracted, and combined the way other parts of a whole can be added, subtracted, and combined. The total number of degrees is the big number. The number of degrees for the two component parts equals the degrees for the whole. Students often have trouble seeing that the total number of degrees in the two smaller angles actually equals the degrees in the larger angle. The color coding shows that this relationship exists. The total for V and P is the total for the white and the gray. The blue angle is also the total for the white and the gray.

If the students have trouble understanding the relationship, have them outline with their finger the entire angle, and each of the smaller angles. Once they get the idea that the total angle is represented two ways (it's both the white and the gray angles combined, and it's the blue angle), they will have a good understanding of the basis for mathematical inferences involving combined angles.

Here's more of the teacher's guide, on what is coming up for us:

Quote

In Level E, students learn the following information about angles:

1) Angles are measured in degrees and are shown with a curved arrow that goes from one line to the other: [figure omitted]

2) The arrow may be close to the point where the lines intersect or it may be farther from this point: [figure omitted]

3) The curved arrow is part of a circle. The number of degrees for a whole circle is 360; the number for half a circle is 180. [figure omitted]

4) The number of degrees for a "corner" is 90: [figure omitted]

5) The same angle is the same portion of a circle, regardless of the spatial orientation. All these angles are 30 degrees: [figures omitted]

6) An angle that is divided into two parts can be shown as the big number in a number family; the two parts are the small numbers. [figure omitted. "Number family" is jargon from the program for a simple notation for a+b = c. The notation puts the three numbers on an arrow, but is possibly similar to what Singapore Math calls a "bar model."]

7) A line intersecting parallel lines creates the same angle at both lines [figure omitted. corresponding angles are congruent]

8 ) Opposite angles formed by intersecting lines are equal. [figure omitted. opposite angles are congruent]

I might be reading "core standards" wrong but I think common core doesn't include this material until 7th grade.

Quote

Students use this information to work a variety of problems.

The first exercises occur in lesson 71. Students learn facts about angles. The angle for the corner of a rectangle is 90 degrees; the angle for half a circle is 180 degrees; the angle for a complete circle is 360 degrees.

Teaching note: Students are to memorize these facts. Students will use the information in later lessons to figure out the number of degrees in fractions of a circle, and will also work problems in which they draw inferences from angles on a "straight line" which forms half a circle (180 degrees)

[figure omitted]

Here's the problem set from lesson 71: 

=======

[figures omitted: four angles labeled r, p, q, and v, opening up in ad hoc directions, of measures 30°, 80°, 118°, and 25°.]

a) Write the degrees for the largest angle

b) How many more degrees is angle p that angle r?

c) How many degrees are in a whole circle?

d) How many degrees are in half a circle?

e) What's the letter for the smallest angle?

=======

In lesson 72, students work with angles that are divided into two parts. Students make a number family to figure out the degrees in one of the angles shown (either one of the smaller angles or the entire angle.

Here's part of the exercise from lesson 72:

=======

[Figures a, b, and c omitted. Each shows a bisected angle, with three angles labeled. Information is shown underneath each figure, for instance figure a is captioned "Angle v is 22°. Angle r is 125°. Figure out angle p."]

d. Problem A. Raise your hand when you know the letters for the two smaller angles that add up to the whole angle.

One smaller angle is V. Everybody, what's the letter for the other smaller angle? (Signal.) P.

What the letter for the whole angle? (Signal.) R.

(Write on the board)

v                          p               r

---------------------->

Here's the number family with three letters.

Your turn. Copy the number family. Then read problem A. Write the two numbers the problem gives and figure out the missing number. Write the answer for item A and box it. Remember the degree symbol. Raise your hand when you're finished. (Observe students and give feedback.)

(Write to show:)

v                          p               r

22                             

---------------------->   125.     103°

The problem tells that angle V is 22 degrees and angle R is 125 degrees. The missing angle is P. Everybody, how many degrees is angle P? (Signal.) 103.

=======

[Sarah0000]

Looks great. If I remember correctly Beast Academy covered this material in the very beginning of level/grade 3. It looks like it's presented in the same way as any addition or subtraction equation so it could be introduced as early as first or second grade level if desired and if basic add/sub is solid, along with combining simple fractions and time segments. 

[UHP]

7 hours ago, Sarah0000 said:

If I remember correctly Beast Academy covered this material in the very beginning of level/grade 3.

I admire BA and have some experience with BA 2 and 3. BA 3 opens with angles, but there might be less there than you remember. The comic has some material on how to tell apart acute, obtuse, and right angles. The concepts are use mainly to classify triangles (right, obtuse, acute). It's noted that a triangle can't have two right angles.

Angle measurements are not treated until early in BA 4 — the little monsters learn to use a protractor.

I'm not sure that "Angles can be added, subtracted, and combined the way other parts of a whole can be added, subtracted, and combined" comes through very clearly in the BA text or the exercises.

"Connecting Math Concepts" introduces this stuff after the students have been working for a long time with fractions and ratios. It discusses angles as "fractions of a circle" and one purpose of the exercises is to give another setting (and more practice) for ratio problems. Here's a problem my daughter worked yesterday (from lesson 79):

 

 

Edited February 23, 2022 by UHP

[Sarah0000]

40 minutes ago, UHP said:

I admire BA and have some experience with BA 2 and 3. BA 3 opens with angles, but there might be less there than you remember. The comic has some material on how to tell apart acute, obtuse, and right angles. The concepts are use mainly to classify triangles (right, obtuse, acute). It's noted that a triangle can't have two right angles.

Angle measurements are not treated until early in BA 4 — the little monsters learn to use a protractor.

I'm not sure that "Angles can be added, subtracted, and combined the way other parts of a whole can be added, subtracted, and combined" comes through very clearly in the BA text or the exercises.

"Connecting Math Concepts" introduces this stuff after the students have been working for a long time with fractions and ratios. It discusses angles as "fractions of a circle" and one purpose of the exercises is to give another setting (and more practice) for ratio problems. Here's a problem my daughter worked yesterday (from lesson 79):

 

 

I'm sure you're right about BA. It's been two years since I had a kid at that level. I'm not sure what you're getting at. Are you asking whether that problem is appropriate for a fourth grader? Or are you saying it's a great program? It looks good to me! So clear and to the point and utilizing multiple mathematical concepts. It reminds me a bit of Singapore with the clear, colorful diagrams already drawn out for the student.

If you're suggesting that this isn't what they are doing in public school, then I would agree with you. I happen to know our local fourth grade class was practicing adding simple fractions with like denominators (seriously, 1/4+1/4). The excuse is that the pandemic has made students behind by two years. 

[UHP]

52 minutes ago, Sarah0000 said:

I'm not sure what you're getting at.

Definitely nothing suspicious! I started the thread because I found an unusual treatment, in timing and content, of angles. I thought I might post followups under it as I put my kid through the lessons, commenting on the experience and comparing it to other things I've tried. But no such updates so far, partly because we are going through them pretty slowly. Once in a while I've gotten life-changing feedback this way. Sometimes crickets.

[mathmarm]

It's much more informative to see the problems from CMC-E. Just the OP was difficult to understand or envision. This definitely goes more advanced than most math curriculum at the K-5 level that I've seen.

 

[mathmarm]

On 2/5/2022 at 9:51 PM, UHP said:

I might be reading "core standards" wrong but I think common core doesn't include this material until 7th grade.

 

On 2/5/2022 at 9:51 PM, UHP said:

In Level E, students learn the following information about angles:

1) Angles are measured in degrees and are shown with a curved arrow that goes from one line to the other: [figure omitted]

2) The arrow may be close to the point where the lines intersect or it may be farther from this point: [figure omitted]

3) The curved arrow is part of a circle. The number of degrees for a whole circle is 360; the number for half a circle is 180. [figure omitted]

4) The number of degrees for a "corner" is 90: [figure omitted]

5) The same angle is the same portion of a circle, regardless of the spatial orientation. All these angles are 30 degrees: [figures omitted]

6) An angle that is divided into two parts can be shown as the big number in a number family; the two parts are the small numbers. [figure omitted. "Number family" is jargon from the program for a simple notation for a+b = c. The notation puts the three numbers on an arrow, but is possibly similar to what Singapore Math calls a "bar model."]

7) A line intersecting parallel lines creates the same angle at both lines [figure omitted. corresponding angles are congruent]

8 ) Opposite angles formed by intersecting lines are equal. [figure omitted. opposite angles are congruent]

I don't know enough about commercial math curriculum to speak with authority, but all of this is included in a high school geometry course, so it's not expected that every student know this before they get to geometry. I'm impressed, but not at all surprised, to see CMC teaching this content at the E-level.

re: #6, unfortunately I have had students arrive to my college classes unaware that there are more than just 4 angle measures. (Acute, Right, Obtuse and Straight) Yes, Number Families are used to show the same relationship as a parts-whole bar model.

re: #7 and #8, are taught in both highschool geometry textbooks that we own. While that may be in some some middle school texts, it it's not always taught in middle school.

Personally, we pull from 2 geometry textbooks while teaching our kids. We are able to organize some of of the material using bar-models which makes it super easy for our kids to pick it up. We do spend a lot of time upfront to make sure that the children can actually read and draw diagrams, as well as take care to teach correct notation as we go along.

Because they're confident with Bar-Models and basic algebraic equations they can follow with the algebra needed to solve all of the problems.

[Clarita]

On 2/23/2022 at 2:46 PM, UHP said:

"Connecting Math Concepts" introduces this stuff after the students have been working for a long time with fractions and ratios. It discusses angles as "fractions of a circle" and one purpose of the exercises is to give another setting (and more practice) for ratio problems. Here's a problem my daughter worked yesterday (from lesson 79):

Interesting, I feel like after "fractions of a circle" I expect the subsequent angles and problems to be in radians instead of degrees.  

[mathmarm]

5 hours ago, Clarita said:

Interesting, I feel like after "fractions of a circle" I expect the subsequent angles and problems to be in radians instead of degrees.  

I wondered about that, but radians are a measure based on the radius of the circle and don't exactly map to a clean fraction of a circle that elementary students will already know.

By teaching angles based on "fractions of a circle", students are able to build onto what they know--a full circle is 360*--and are enabled to use skills that they know--rates, ratios and proportions ( or fraction multiplication <-> division) to solve problems at their current level. This approach enables students to intelligently and reliably tackle 2 and 3 step problems at their level from day one.

Ultimately, the goal is to instruct and develop the students ability to work more meaningfully with what the know, and make connections between what they're learning and relate that back to what they've learned. Too often, students are required to work only with supplementary and complementary angles prior to geometry--this is for students 6-8 who only have to add/subtract from 180 or 90 degrees.

Here, the 4th grade student must connect multiple steps

1) M + R = Q
2) R is 1/20 * 360 or 360/20, so 18
3) M + R is Q, so 40 + 18 = 58

@UHP I'm interested to see how they develop this idea in later levels.

[UHP]

9 hours ago, Clarita said:

Interesting, I feel like after "fractions of a circle" I expect the subsequent angles and problems to be in radians instead of degrees.  

Well, you are more likely to get a whole number after multiplying a fraction by 360, than by twice pi. I'd like to speculate that this is how the "degrees" measurement caught on in history: people who were innocent of fractions but comfortable with whole numbers could still do some figuring with angles.

Radians are more suited to figuring out lengths, areas, and volumes. In parallel to these angle exercises, CMC-E (or my 1990s edition of it) has been covering simple area and perimeter problems: finding the perimeter of a parallelogram, given two of its side lengths. Or find the area given a base length and height. Don't confuse height of a parallelogram for a side length. Don't neglect the unit name.

Area and perimeter of a circle come up after (I think) lesson 90 in this program, we are on lesson 82.

I'm not sure that the program uses the word "radian." The students do work many problems like this:

Quote

"the ratio of dogs to cats is 3 to 8. If there are 16 cats then how many dogs are there?"

I haven't thought it through but perhaps radians could be introduced with a similar problem format:

Quote

"Here's the rule for radians: The ratio of radians to degrees is about 3.14 to 180. If there are 90 degrees then about how many radians?"

Two obstacles to doing this: that word "about," and decimal division. I don't know how significant these obstacles are.

[UHP]

3 hours ago, mathmarm said:

@UHP I'm interested to see how they develop this idea in later levels.

I haven't totally decoded the teacher's guide for "Level F" yet, but it looks like angles are mostly reviewed, and not put to new uses or developed further. Too bad!

[Clarita]

11 hours ago, mathmarm said:

I wondered about that, but radians are a measure based on the radius of the circle and don't exactly map to a clean fraction of a circle that elementary students will already know.

I do suspect the pi part is what keeps them from teaching radians immediately after introducing the fraction of a circle. Actually, if you just assume pi to be constant, radians are very clean fractions of a circle. 2*pi represents a complete unit circle so looking at radians it's easier than degrees to figure out exactly where you are in a circle. 

7 hours ago, UHP said:

Radians are more suited to figuring out lengths, areas, and volumes.

Radians are super powerful to solve periodic (repetitive) things because of how easily it maps on a circle. Fascinating application is noise cancellation, auto-tune or in general manipulation of sound.  I know it's mainstream and commonplace to introduce radians much later in math (middle school/high school). Even then, it's usually quite vague as to why we are learning this piece of information at all. Since I've learned that kids can learn about placeholders pretty early in math, it just seems like it would be a great little nugget to throw in there as why we would have 2 different angle measurements.

In terms of introduction, probably have to go through the introduction of the circumference of a circle is 2*pi*r (that I think is complicated, but I suppose they could just memorize it as fact until polar coordinates and integrals). You can physically approximately show the 3.14 ratio occurring by doing an actual measurement of a circle or area of a sphere. Then imagine you have a circular track with a radius of 1, you can define each point on that track using radians. Values above 2*pi just means you've gone around several times. 

Then with the understanding of what degrees and radians with relation to a circle how to convert between the two becomes obvious and if a student needs an explicit way of doing it show the procedure, otherwise visually it's a map of both on a circle (one on the circular path and the other as slices).

[UHP]

On 3/1/2022 at 11:06 AM, UHP said:

Area and perimeter of a circle come up after (I think) lesson 90 in this program, we are on lesson 82.

We reached lesson 91 today. I haven't started it. The program has the student use a calculator to illustrate pi. It's elegant in its way but my daughter so far hasn't learned to use a calculator and for now I prefer to keep it that way. I'll think about how to do it without a calculator.

The lesson opens with some information about circumference and diameter:

"You're going to learn about circles. I'll read what it says. Follow along..." and afterwards three guided exercises I've copied below.

Some twenty lessons ago the program had me train my pupil to solve oral questions like "7 times some fraction equals 22. What fraction?" The answer is "twenty-two sevenths." She also knows how to figure out the "mixed number" (3 and 1/7) that equals 22/7, but doesn't yet know how to crank out 22/7 in decimal form. She did learn in the previous lesson how to round decimals (after practicing rounding whole numbers for a few lessons before that).

The program asks you not to read out 3.142 as "three point one four two" but as "three and 142 thousandths." Average human body temperature isn't "ninety-eight point six" but "ninety eight and six tenths." But I haven't always been able to stop myself.

Quote

The diameter and the circumference of any circle are related by multiplication. That means you can multiply the diameter by a number to get the circumference. The table shows the diameter and the circumference of different circles. You're going to figure out the fraction you multiply by.

* Circle A. How many inches is the diameter? (Signal.) 7.

How many inches is the circumference? (Signal.) 22.

So you'll work the problem 7 times some fraction equals 22. Your turn: write the equation with the missing fraction. Raise your hand when you're finished.

(Write on the board:) (a) 7 (22/7) = 22

Here's what you should have. The fraction is 22/7. Raise your hand if you got it right.

Use your calculator. Divide 22 by 7 and round the answer to two decimal places. Raise your hand when you're finished.

Everybody, what's 22 divided by 7? (Signal.) 3 and 14 hundredths.

* Circle B. How many centimeters is the diameter? (Signal.) 21.

How many centimeters is the circumference? (Signal.) 66.

Write the multiplication equation that starts with 21. Raise your hand when you're finished.  (Observe students and give feedback.)

The equation you should have is 21 times 66/21 equals 66. Use your calculator and figure out the decimal value for the fraction. Round the number to two decimal places. Raise your hand when you've written the decimal number for circle B.

Everybody, what's the decimal number? 3 and 14 hundredths.

That's the same number you got for circle A.

* Circle C. How many meters is the diameter? (Signal.) 5 and 1-tenth.

How many meters is the circumference? (Signal.) 16.

Write the equation and show the fraction you multiply by. Raise your hand when you've done that much.

(Write on the board:) (c) 5.1 (16/5.1) = 16.

Here's what you should have. When you divide 16 by 5 and 1/10, you must enter the decimal point -- 5, decimal point, and 1. Do the division and figure out the decimal number rounded to two decimal places. Raise your hand when you're finished. (Observe students and give feedback.)

Everybody, what does that fraction equal? (Signal.) 3 and 14-hundredths.

That's the same number again.

* For each circle, your fraction shows the circumference divided by the diameter. It's always the same number. Everybody, what's that number? (Signal.) 3 and 14-hundredths.

* Isn't it amazing that you'll always get that number when you divide the circumference by the diameter?

Remember that number. You'll use it a lot.

 

[daijobu]

The point about the ration of circumference to diameter might hit harder if you measured circular items of all sizes around your house.  Measure using a string the circumference and diameter of a can, a round plate, a bicycle wheel, anything you can find.  You can use a calculator together or even long division to find that the ratio is always approximately 3.  I think that's more effective than having the numbers presented to you in a worksheet.     

[Clarita]

I watched the "History of Maths" documentary he showed area, circumference and diameter in terms of marbles in a circular container. Even as an adult I found that fascinating and a great illustration, I think he used 3 and a little bit more. He was demonstrating how ancient people could have had a sense of how to make and measure round things before decimals were invented. 

[mathmarm]

9 hours ago, UHP said:

We reached lesson 91 today. I haven't started it. The program has the student use a calculator to illustrate pi. It's elegant in its way but my daughter so far hasn't learned to use a calculator and for now I prefer to keep it that way. I'll think about how to do it without a calculator.

The lesson opens with some information about circumference and diameter:

"You're going to learn about circles. I'll read what it says. Follow along..." and afterwards three guided exercises I've copied below.

If she doesn't already know them, I'd pause the program for a couple of a days and teach the parts of a circle using cardboard cut-outs and markers.
Once she knows the the circumference, radius and diameter of a circle, you can explore circles around the house (bowls, cubs, rolls of tape, bottle caps, tires, etc).

We use a ribbon (or Yarn) and a marker. We measure and mark the diameter of a circle on a ribbon, then, wrap it around the edge (circumference) and clip the ribbon/string. We have the kids fold the ribbon to see see how many times longer the circle is than it's diameter.

It doesn't take them long to notice that the circles circumference is always a little more than 3 times as long as the diameter.

9 hours ago, UHP said:

Some twenty lessons ago the program had me train my pupil to solve oral questions like "7 times some fraction equals 22. What fraction?" The answer is "twenty-two sevenths." She also knows how to figure out the "mixed number" (3 and 1/7) that equals 22/7, but doesn't yet know how to crank out 22/7 in decimal form. She did learn in the previous lesson how to round decimals (after practicing rounding whole numbers for a few lessons before that).

As for computing sans calculator. I suggest that you use place-value units and the distributive property, so for 22/7, my kids are taught to work it as follows:

Rewrite with the division-house: 7 ) 22

Next decompose 22 into multiples of 7.

I tell them which place value I want them to go out to. So if I tell my kids to do 22/7 to the ten-thousandths place they will do it like this.

7 ) 21 + 1.00

7 ) 21 + 0.70 + 0.28 + 0.020

7) 21 + 0.70 + 0.28 + 0.014 + 0.0056 + ...

Then, they just divide each chunk by 7 and get 3+0.1+0.04 + 0.002 + 0.0008 or 3.1428