This unit follows naturally from the Smart Doubling unit.
In this unit students are encouraged to further develop part/whole mental methods by using the strategy of "make a ten".
Specific Learning Outcomes
- Demonstrate automatic recall of all pairs of single digit addition facts whose total is 10 or less.
- Use the mental strategy "make a ten" for addition problems.
- Use the most efficient mental strategy for a given problem.
Description of Mathematics
In this unit students are encouraged to add to their use of part/whole with doubles by using make a ten methods.
Examples of part/whole methods using make a ten:
The students work out 8 + 5 by removing 2 from the 5 to leave 3, add this 2 to the 8 to make 10 then add the 3 to the 10 to give 13.
The students work out 38 + 8 by removing 2 from the 8 to leave 6, add the 2 to the 38 to give 40, then add 6 to the 40 to give 46.
It is desirable for students to move to part/whole methods as counting methods fail for larger numbers. For example, a student who attempts to work out 36 + 46 by counting on will soon lose their way, whereas the part/whole thinker could solve this by adding 30 and 40 to give 70, then add 6 and 6 to get 12 then add 12 to 70 to get 82.
Students who have successfully understood the part/whole methods in the Smart Doubling unit will have little trouble learning the make a ten part/whole strategy. Teachers should expect those students who failed to make the part/whole connections with doubles to also struggle with this unit. It may be best not to introduce this unit until the students understand the doubling strategy.
Required Resource Materials
- Sets of counters
- Empty tens frames with squares large enough to contain counters
- Tens frames (Material Master 4-6)
- Check the students’ knowledge of the addition facts that add to ten facts before starting. This unit builds on the students’ knowledge of the known facts of ten, i.e., 1 + 9 = 10, 2 + 8 = 10, 3 + 7 = 10, 4 + 6 = 10, 5 + 5 etc.
- Also check that the students understand the " teen" numbers. For example, show the students a ten and a four on tens frame as shown. They should respond that 10 + 4 is 14 without needing to count-on by ones.
- Pose an addition problem where one of the numbers is just under 10:
Anaru has 9 pink lollies and 6 yellow lollies. How many lollies does he have altogether?
Ask the students to model this with counters on tens frames.
- Ask the students to think about ways that they could make this problem quick (or easy) to solve. Encourage the students to think of ways that do not involve counting on by ones.
For example, Anaru moves a yellow lolly to the pink lollies. Now ten and five is shown.
It is important to discuss the fact that the answer to 9 + 6 is the same as the answer to 10 + 5. Without this realisation students will not be able to make progress into mental number processes. Although the problem can of course be correctly solved by counting-on the aim of the lesson is to encourage the development of part/whole mental strategies.
Over the next 3-4 days the students are encouraged to use make to ten strategies for solving number story problems.
- Make up number stories for 9 + 7, 5 + 8, 9 + 8, 7 + 6. Students use counters on empty tens frames and use the technique of filling one of the frames up to show ten counters.
- Students now use the pre-printed tens frames.
Moana has 7 oranges and 9 apples. How many pieces of fruit does she have altogether?
Ask the students to show 7 + 9 with a pre-printed 9 card and a pre-printed 7 card. Students visualise moving 1 out of the 7 to leave 6, and adding 1 to the 9 to create 10. This action of imaging pushes the student towards understanding the movement of numbers around within a problem as a key to mental processing.
- Next, only you, as the teacher, have the pre-printed tens frames. Hold the cards so that only you can see the dots and the students have to visualise what you are seeing. Prior to giving the students addition problems ask the students to first visualise single numbers.
I can see the 8 card. What does it look like?
Some students will "see" the 5 + 3, others will "see" the two spaces.
- Next pose addition problems, which encourage the students to visualise the tens frames that only you can see. For example:
I can see 8+ 6. How could I move the dots to work out 8 + 6?
- The next step in the progression involves no materials. The students visualise the tens frames to move around dots as they solve problems:
8 + 5, 3 + 9, 8 + 9.
- Now link the doubles to the make a ten strategy. Ask the students to consider all the ways they know to work out 8 + 9. The range of likely responses is:
- Double 8 + 1 = 16 + 1 = 17
- Double 9 – 1 = 18 – 1 = 17
- Double 10 – 1 – 2 = 20 – 1 – 2 = 19 – 2 = 17 (Rarely used)
- 10 + 8 = 18 but this is 1 too many so the answer is 17
- Remove 1 from the 8 and add 1 to the 9 and to give 7 + 10 which is 17
- Add 2 to the 8 and remove 2 from the 9 and to give 10 + 7 which is 17
After working through this example together students work in pairs to solve: 6 + 8, 9 + 7, 6 + 9. They discuss their strategies.
- Attention now turns to using make a ten strategies to solve subtraction problems. Subtraction problems with the first number in the teen decade are given.
Michael has 13 lollies and he eats 5 of them. How many lollies does Michael now have now?
Students model 13 on pre-printed ten frames.
Two equally good methods are likely.
EITHER remove the 3 to leave the 10 then remove a further 2 to leave 8.
OR remove 5 from the 10 to leave 5 and add the 3 to give 8.
Students use pre-printed tens frames to solve number stories for subtraction problems. Use: 14 - 6, 17 - 8, 12 - 8, 17 - 5, 13 - 5, 13 - 7, 14 - 3.
Notice the presence of subtractions like 17 - 5. They prevent the mindset that may develop that make a ten is always appropriate. In fact, in this example, breaking the ten is not necessary because 7 - 5 = 2 and add the 10 back in gives 12.
- Now work on using a strategy than involves solving subtraction problems by adding.
What is 13 - 9?
Charlotte thinks of a way to work out 13 - 9 by adding. She asks "9 and what makes 13?" and comes up with the answer 4.
Ask the students how they think Charlotte does this. (She has gone 9 + 1= 10, she remembers the 1, she goes 10 + 3 = 13 and adds the 1 to the 3 to get 4.)
Have the students use Charlotte’s addition method to work out these problems: 12 – 9, 14 – 8, 17 – 14, 16 - 7
- Have students explore both ways they have learnt to solve subtraction problems by using either an addition and a subtraction way to work these problems: 13 - 9, 13 - 4, 17 - 15, 17 - 2, 19 - 17 , 12 – 8.
At the end of each session encourage the students to share and discuss answers as a group or class.
Dear family and whānau
At school this week we have solved addition and subtraction problems using a strategy called "Making Tens". You can help your child by asking them to explain their thinking as they solve the problems on this sheet.
To work out 9 + 7 Moana removes 1 from the 7 to leave 6 and adds this 1 to the 9 to make 10. She then knows 9 + 7 = 10 + 6 so the answer is 16.
Use Moana’s method of making a ten to work out these:
To work out 9 + 8 William knows a number ways.
- His first way is 9 + 9 = 18 and remove 1 to give 17
- His second way is to remove 1 from the 9 to leave 8 then add 8 and 8 to give 16, then put the 1 back to give 17.
- His third way is to remove 1 from the 8 to leave 7, then add that 1 onto the 9 to make 10. Adding the 10 and the 7 makes 17.
Write stories to show two different ways to work out the following:
9 + 7
15 - 9