How to explain this to a 3rd grader???

Question

Trying to explain this to my 3rd grader without algebra. No teacher help- so frustrated. This is a test question (we are reviewing what she got wrong) from Math Expressions unit 6. Do I just say make it a square and divide by 4, and then multiply the length x width? I have no idea how the teacher taught it and no idea how this child would explain how she got the answer.This question seems insane for 3rd grade!!!

A farmer has 80 feet of fencing to fence four sides of a rectangular garden. What is the greatest area of garden that the farmer can fence? Explain how you know.

Please help me- I want her to understand this, but I need a simpler explanation on a 3rd grade level.

Answer 1

I would let her discover the answer. Let her start with a rectangle and continue to make shorter side of rectangle smaller while making larger size smaller and find area.

Ex

5x35 rectangle has area of 175

10x30 rectangle has area of 300

15x25 rectangle has area of 375

20x20 rectangle has area of 400

25x15 rectangle has area of 375

Etc

Answer 2

Hi Katherine,

I would begin by drawing a diagram. Draw a rectangle and explain that the opposite sides of a rectangle have the same length or measure and that one horizontal side is called the length and one vertical side is called the width. This is not beyond the comprehension of a third grader. Having a diagram to look at helps in understanding and a third grader can clearly see that the opposite sides of a rectangle, if drawn neatly with a ruler, appear to have the same length. Your daughter can also measure the opposite sides herself with the ruler. Now explain that the lengths of all of the four sides added together must equal 80 feet because that is how much fencing the farmer has. Again, stress that the opposite sides have the same length. You can also explain that one length and one width added together must equal 40 feet, half of 80 feet. Now you have to use trial and error. Choose two numbers, one to represent the length and one to represent the width, that add up to 40 feet. Because 80 is a multiple of 10 and a multiple of 5, you can suggest that she use multiples of 5 or multiples of 10. Now see what your daughter choses for numbers and guide her and correct her if she makes any arithmetic mistakes. From here on you increase one number by 5 (or 10) and decrease the other number by 5 (or 10). You would now have to explain how to find the area of a rectangle. Area=length x width or A=L x W. Your daughter will soon discover a pattern whereby as the smaller number increases and the larger number decreases, the smaller number now becomes the larger number and the larger number now becomes the smaller number and you now have repetition of numbers.

You find the area each time you change numbers until your daughter realizes that 20 x 20 will give the largest area. This is true in mathematics; a square has the largest area. Doing the problem this way gives your daughter practice in adding and multiplying as well as patterns.

5 x 35

10 x 30

15 x 25

20 x 20

25 x 15

30 x 10

35 x 5

After you come to the conclusion that 20 x 20 gives the largest area, you can try other numbers that are not multiples of either 5 or 10. You could use 8 and 32, 12 and 28, 16 and 24, and whatever other numbers you like.

The algebra for this type of problem is as follows:

P=2l+2w and A=lw

80=2l+2w, 40=l+w, 40-l=w (or 40-w=l)

A=lw, A=l(40-l), A=40l -l^2, A=-l^2+40l

The maximizing point is h=-b/2a where a=the coefficient of l^2 which is -1 and b is the coefficient of l which is 40. Therefore h=-40/2(-1), h=-40/(-2), h=20

A=lw, A=20(20)=400 square feet. Each side of the rectangle must be 20 feet so you have a 20 by 20 square.

Answer 3

I totally get how frustrating this can be! Here’s a simple way to explain it without algebra, using hands-on thinking that a 3rd grader can understand.

**Step 1: Start with the Perimeter Concept**

Tell your child:

*"We have 80 feet of fence, and we need to use all of it to make a rectangle. That means the total length of all four sides must add up to 80 feet."*

**Step 2: Try Different Rectangles**

Instead of jumping to a square, encourage your child to think about different rectangles they can make with 80 feet of fencing. For example:

- If the garden is **10 feet wide**, then the two widths take up **10 + 10 = 20 feet**. That leaves **60 feet** for the two lengths, so each length would be **30 feet** (since 60 ÷ 2 = 30).

- If the garden is **20 feet wide**, then the two widths take up **20 + 20 = 40 feet**, leaving **40 feet** for the two lengths, so each length would be **20 feet** (since 40 ÷ 2 = 20).

**Step 3: Find the Area**

Explain:

*"The area is how much space is inside the garden. We find the area by multiplying the length times the width."*

- **10 x 30 = 300 square feet**

- **20 x 20 = 400 square feet**

**Step 4: Find the Biggest One**

Ask your child:

*"Which one is bigger, 300 or 400?"*

Since **400 is bigger**, the best choice is **a 20 ft by 20 ft square garden**.

**How to Explain Why a Square Works Best**

If they ask why, you can say:

*"A square spreads out the fencing evenly, so we get the most space inside. If we make the garden too long and skinny, we lose space."*

You can even draw different rectangles and count squares inside to show which one holds the most.

Would this explanation help your daughter understand? 😊