Can I split static IP range between 2 routers, with an upstream switch?

Question

My ISP provides static IP range x.y.z.112 to 127. At the moment, my existing router is configured with

ROUTER A (existing settings)

• WAN IP: x.y.z.114
• Mask: x.y.z.240
• Gateway: x.y.z.113

I want to split the IP range such that x.y.z.120 to 127 goes through another router. To do this, my understanding is to change the above mask to 255.255.255.248 and use on the new router the settings:

ROUTER A

• WAN IP: x.y.z.114
• Mask: x.y.z.248
• Gateway: x.y.z.113

ROUTER B

• WAN IP: x.y.z.120
• Mask: x.y.z.248
• Gateway: x.y.z.113

and then add a switch upstream of the 2 routers.

However, just by changing the subnet mask of the existing router alone, the internet connection is lost. What am I not understanding or doing correctly?

Desired Network Diagram

ISP

• WAN Gateway: x.y.z.113 (provided by ISP)
  • Switch
    • Router A: WAN: x.y.z.112-119
    • Router B: WAN: x.y.z.120-127

Answer 1

In your first (working) configuration you have the gateway (.113) on the same subnet. When you change the mask to .248 and the last octet to .120, you "push" the gateway out of the ip-range of the network.
The new configuration looks as follows:
Network name: x.y.z.120
Host-range: x.y.z.121-126
Broadcast: x.y.z.127
As you can see, the gateway .113 is not in the same subnet. In short, you no longer have a valid default gateway. Without a GW, there is no way network traffic is going to find its way beyond its own subnet.
With the information you give, it seems that you are trying to get more public addresses than your ISP provides. If you would supply a simple drawing of what you had in mind it would be alot easier to aid you.

Update - new data from OP:
It will not work the way you like to have it. The thing is you are trying to make two subnets go through the same GW without a router and the technology of IP-subnets just don't work that way. Already in your edited post you can see that .113 falls outside the IP-range of router B.
One option could be to connect two routers in serie:

ISP GW (.113)
|
Router A (IP-range: .112-.119)
|
Router B (IP-range: .120-.127)

This would simply mean that traffic from router B goes through router A and on to the ISP. In that way you get two separate networks on the same GW. I must however ask why you want two subnets?