5.G.A.2 Gates and Fences (Linear Functions)

5.G.A.2   Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.

The goal of this lesson is to make sense of the  y = mx + b formula. The children create a T-chart and record the gate starting point, plus the increasing pattern of identical fence sections.

My students love "playing" with the pattern blocks. In this activity, I told them to design at least two identical fancy fence sections and one gate. In order to keep the gate and fence numbers reasonable for the mathematics, I limited the size of each to be under 12 blocks (if possible). Sometimes the design is so creative that I give in a little.

 Kimberly's pattern has 8 blocks in the gate, plus 6 blocks in each fence section. Her formula is  y = 6x + 8. If she orders 10 fence sections and a gate, then she will have 10 x 6 blocks in each fence section plus eight blocks in the gate for a total of 68 blocks needed to construct her fence.

Ediverto, a very bright child with autism, loves writing the numbers to prove his gate and fence formula.



Both his fence and gate parts each have 4 blocks. (His drawing is not clear, but he used 1 red block and 3 blue blocks to construct his fence.) So  y = 4x + 4.

Fatima determines her x and y ordered pairs and graphs her fence. She sees a pattern of going up on the y-axis by 7 and then over one each time. This helps her understand that she is multiplying by 7 as she continues to add new fence parts.  She started at (0,4) on the y-axis and went up 7 each time and over one. This helped her see why the "+ b" portion of the linear formula is also called the y-intercept. I helped her make the connection with how she started first with her gate of 4 blocks and then kept adding the fence sections.

The students enjoy working together in table groups to compare answers, justify their formulas, and make sense of the fence problem.

They determine how to scale their y-axis and  complete their graphs.  I suggested counting by ones or twos on the y-axis, but the children are allowed to change this if their block numbers are larger.








Later in the week, Leslie shares her solution to the fence problem. She explains to the other students why her formula is y = 8x + 10.

The more time we spend reviewing each fence and gate pattern, the deeper their understanding grows. Besides, each child loves coming up to the front and being the star of the show.  :)

How many formulas can you find?

Jason has a pattern of 9 in his gate plus 3 in each fence part.

What is his formula?  What does y equal?

Cesar has 10 blocks in his gate and 4 in his fence pattern before it starts to repeat again.

Galilea has a gate with 12 blocks and a fence part with 8 blocks.

Do you know what y equals?

(I'm sure you do by now!)

             y = 8x + 12

                                 Good job, Gali!

Martha, Pablo, and Alexis had fun making their fence patterns stand up like real fences.

Johnny's fence section has 4 blocks (two orange and two greens) before it repeats again.










Jonathan's bird gate is so creative that I had to allow it...and he's very bright, so the numbers are okay for him to work with.

A great introduction to next years 5th grade standards! And a lot of fun for everyone.