Structuring Numbers using the Arithmetic Rack

The Arithmetic Rack allows for a variety of solution strategies The Arithmetic Rack allows for a variety of solution strategies. Students are able to make use of the number relations they can think of. These activities induce a shift in children’s thinking from visual, or tacitly countable items to an increasingly numerical interpretation.

│----------------------- ОООООООООО│ The beads that are shifted to the left are the ones that count. The beads shifted to the right don’t.

Students may think of setting up a given number in different ways Students may think of setting up a given number in different ways. Here are some possibilities for the number 7. │ООО-----------------------ООООООО│ │ОООО-----------------------ОООООО│ Students may think of 7 as 3 + 4, or as 6 + 1 = 3 + 3 + 1.

│ООООО-----------------------ООООО│ │ОО-----------------------ОООООООО│ Students may think of 7 as 5 + 2.

│ООООООО-----------------------ООО│ │-----------------------ОООООООООО│ Students may think of 7 as 7 ones, 5 + 2, or as 10 – 3.

│ОООООО------------------ОООО│ │О------------------ООООООООО│ This does show the number 7; however, the previous ways are encouraged over this way because it does not reference 5’s or doubles.

Determining the total number of beads │ООООООО-----------------------ООО│ │ОООООООО-----------------------ОО│ This may be seen as 7 + 8, however the students have to figure out how much that is. The same configuration may be thought of as 5 + 2 and 5 + 3, which can be totaled as 5 + 5 = 10; 2 + 3 = 5; 10 + 5 = 15. Or, it can be seen as (7 + 7) + 1 = 15.

To help students show one quantity: Show a number (any way) without a discussion of efficiency. Show a number with a discussion of efficiency. Does the showing make it easy or hard to ‘read’ the number of beads? Without counting, can you recognize a quantity? Anticipate a quantity?

The Arithmetic Rack can be used as a way to keep track of the number of people on the upper and lower decks of a double-decker bus. Six people are sitting on a bus. Three are on top and 3 are on the bottom. At the next stop, 9 more passengers get on. How many people are now riding on the bus?

│ООО-----------------------ООООООО│ Nine could be added as 9 = 5 + 4. │ООО-ООООО-----------------------ОО│ │ООО-ОООО-----------------------ООО│ 10 + 5 = 15

Nine could be added as 7 + 2 (filling up the upper 10 first). │ООО-ООООООО-----------------------│ │ООО-ОО-----------------------ООООО│ 10 + 5 = 15

Nine could be added as 8 + 1, with 8 = 4 + 4. │ООО-ОООО-О----------------------ОО│ │ООО-ОООО-----------------------ООО│

Staggered Tasks The teacher gives the first number, and the students make it on their AR. Then the teacher gives the second number (using the story of the bus). Students complete the problem on their AR’s. Discuss the results.

Anticipating the second step. The teacher shows a first number. Students make the same number in the same way on their AR’s. The teacher gives the second part of the problem but BEFORE students complete it on their AR’s and discuss how they plan to do so.

Reasoning without moving the beads. The teacher shows a number and asks questions such as how many people need to get on (or off) to get another number of people on the bus.

Imagination Bingo The teacher gives the first number that students make on their AR’s. Then they IMAGINE completing the problem and show the result on their Bingo cards.

Anticipating Efficient Ways The teacher poses the entire problem (7 people are on the bus and 8 more get on). BEFORE students solve the problem on their AR’s they discuss how they plan to do it and why they have the plan that they do.

Number Sentences without the Rack The teacher presents written number sentences which students are to solve without using their AR’s. We want the students to use rack-related strategies. They may talk in terms of moving the beads on the rack.