In this unit we work on word problems about climbing up steps and riding in a lift. We learn about different types of problems that can be modelled on number lines. We practice mental calculations with sums to 10 and to 20.
- Use the number line to model their solutions.
- Use mental strategies to solve problems.
- Explain their mathematical thinking in solving problems.
The number line is a useful concept in mathematics. In the early stages of number it can be used to get an idea of the progression of numbers and help students to see the relative positions that numbers hold. It can then be used to solve simple arithmetic problems.
This unit is based around the use of number lines. Although we tend to think of number lines as being horizontal, there are advantages in thinking of them as being vertical. In this orientation they become more obvious tools for solving problems in the vertical plane such as going in lifts and walking up stairs. Later they are useful for undertsanding scales such as temperature on a vertical themometer.
In later work at higher levels, both horizontal and vertical number lines are used together. Working together they provide the very powerful tool of cartesian coordinates. This device enables us to ‘picture’ algebraic quantities.
In this unit, there are three different types of problems that the students should be encouraged to work with in this unit. These types are where (1) the result is unknown; (2) the change is unknown; and (3) the start is unknown. They are not necessarily related to vertical number lines but we will use them that way in this unit. We now discuss them in more detail.
(1) The result is unknown. In this type of problem two numbers are given and the students have to find the result. (For example, 4 + 6 = ?. ) In a practical situation we might have a problem like this:
Tim takes the lift to see his mother at work. It stops on the way at the fourth floor. Then it travels up 6 more floors. Which floor does Tim’s mum work on?
Students should be encouraged to make up other stories using this type of problem structure of possible trips in a vertical direction and learn to model them on a vertical number line. Another problem could be:
Jo climbed up a ladder. First she stopped on the third rung. Then she went up four more rungs. Which rung did she stop on?
(2) The change is unknown. In this type of problem an initial and a final number are given and the students have to find the number in between. (For example, 5 + ? = 10.) In a practical situation we might have a problem like this:
Tim takes the lift to see his mother at work. Tim’s mum works on the tenth floor. If the lift stops at the fifth floor, how many more floors does Tim have to travel?
This problem continues to use the addition facts with sums to 10 but in this case one of the addends is missing. Students should be encouraged to make up other stories using this type of problem structure of possible trips in a vertical direction and learn to model them on a vertical number line. Another problem might be:
Jo climbed up a ladder. She wanted to get to the tenth rung. After a while she got to the third rung. How many rungs did she have to go?
(3) The start is unknown. In this type of problem we know what has happened to an unknown number to give a particular answer. We have to find the unknown number. (For example, ? – 5 = 2.) In a practical situation we might have a problem like this:
Jenny leaves her mother’s work to go down in the lift. She travels down five floors before the lift stops at the second floor. On what floor did she start?
This problem continues to use the addition facts with sums to 10 but in this case the starting addend is unknown. Students should be encouraged to make up other stories using this type of problem structure and learn to model them on a vertical number line. Another problem might be:
Jenny leaves her mother’s work to go down the steps. She walks down six steps and stops four steps from the ground. On what step did she start?
- A painted number line(s) in the playground.
- A ladder in the school playground.
- Number lines (both with all numbers and empty, using strips of paper with paper clips, clothes pegs on a string, or drawn number lines).
We introduce the session by asking the students to move backwards and forwards on a painted (or chalk drawn) number line in the playground.
- Begin the session by getting the students to stand on any number on the number line and asking them which direction will they move to a larger number and which direction for a smaller number. Which direction will you move for the next largest number? Now move there.
Which direction will you move for the next smallest number? Now move there.
Where are you now?
Ask the students to take a different number as a starting point. Repeat a couple times varying the instruction to include moving to the numbers that are 2, 3, 4, or 5 more and less than where they started.
- Pose a problem where the result is unknown. For example, Jenny wrote the first 2 pages of her story before the bell rang. After playtime, she quickly wrote 4 more pages of her exciting story. After talking to her teacher about it, Jenny wrote 1 more page. How many pages long was the story?
Can you use the empty number line to work out how many pages long Jenny’s story is?
What numbers will you need on your number line to show how Jenny wrote her story?
Which direction will you jump? Why? How will you show this?
- Ask the students to make up their own problem and give it to another student to first estimate the answer and then use an empty number line to find the answer. Ask the students to explain how they solved the problem. Are there other ways of solving it?
Over the next 2 to 3 days the students pose a number of problems for each other. They are encouraged to model their solutions on an empty (or filled) number line and explain their answers to others. They will begin to think about the most efficient ways of solving the problems. It is important that students are provided with opportunities to build up addition facts to 10 and to 20. Some students may solve these problems without the number line.
- Begin by encouraging the students to think of situations where they might use a vertical number line to show how they solved an addition problem with sums to 10. This gives students experience of a different orientation. If possible provide an opportunity for students to experience a situation that involves vertical movement. This might include riding an escalator, or climbing a ladder or climbing steps in a playground. If this is not appropriate, read a story or look at a picture or video of such an experience.
- Ladder problems.
The ladder on the slide has ten steps. Sue stops on the fourth step to have a rest. How many more steps has she got to climb to get to the top?
Ask the students to explain their answers and to model these on a number line.
Get them to suggest other places where Sue might stop on her way to the top of the slide. Ask them to model these on a number line.
- Lift problems
Pose problems based on riding in a lift in a tall building such as an office block. In this sequence include problems with addition facts with sums to 10. It is important to pose different types of word problems. (See, A description of the mathematics explored in the unit, above.)
- Other problems to select from:
The next day Tane takes the lift to see his mother again. This time the lift stops twice on the way to the tenth floor so different people can get in and out. If one of these stops is the fifth floor what other floors might the lift stop at? Show me on a number line all the places where the lift might stop?
Ask the students to record all the possibilities of places the lift stops on a vertical number line
Note: It is important to build up the idea of five as useful benchmark in addition. The problem above provides an opportunity for the students to practice number facts to 5.
- Extend the problems to include addition facts to 20.
Tane takes the hotel lift to see his cousins from Australia. It stops on the way at the tenth floor. Then it travels up 6 more floors and stops and then it travels another 4 floors. Which floor are Tane’s cousins staying on?
This problem uses the addition facts of 10 and 6 and 4. Students should be encouraged to make up other stories of possible trips in the lift to Tane’s cousins’ room and share these with a partner.
The next day Tane takes the lift to see his cousins again. This time the lift stops twice on the way to the twentieth floor so different people can get in or out. If one of these stops is the tenth floor what other floors might the lift stop at? Show me on a number line all the places where the lift might stop?
Note: It is important to build up the idea of ten as useful benchmark in addition. The problem above provides an opportunity for the students to practice addition facts of 10 and what make 20.
Use other types of word problems such change unknown and start unknown (as explained above) for Tim’s trip in the hotel lift.
On the final day of the unit ask the students to invent some different stories that can be modelled on a vertical number line using sums to 20. They should include all three problem types. Share the stories with others in the class.